Lakshmi Star

Lakshmi Star

Lakshmi star is an octagram , a composite regular star-shaped polygon represented by the Shlefli symbol a {8}, {8/2} or 2 {4}, made up of two squares with a common center rotated by 45 ° relative to each other ^[1] . In Hinduism it is a symbol of ^[1] , the eight forms of the goddess Lakshmi .

Relationships

The length of side a of the star can determine the length of side b of the inner octagonal intersection of two squares ^[1] :

$b=a({\sqrt {2}}-1).$ ${\ displaystyle b = a ({\ sqrt {2}} - 1).}$

The areas of union and intersection of two squares are calculated by the formulas

$S_{\cup }=2(2-{\sqrt {2}})a^{2},$ ${\ displaystyle S _ {\ cup} = 2 (2 - {\ sqrt {2}}) a ^ {2},}$

$S_{\cap }=2({\sqrt {2}}-1)a^{2}.$ ${\ displaystyle S _ {\ cap} = 2 ({\ sqrt {2}} - 1) a ^ {2}.}$

A star can be inscribed in a circle with a radius

$R={\frac {\sqrt {2}}{2}}a.$ ${\ displaystyle R = {\ frac {\ sqrt {2}} {2}} a.}$

In the inner octagon, you can enter a circle with a radius of half the side of the star. If you connect the tops of the star, a new octagon is formed with the side and area

$l=a{\sqrt {1-{\frac {\sqrt {2}}{2}}}},S={\sqrt {2}}a^{2}.$ ${\ displaystyle l = a {\ sqrt {1 - {\ frac {\ sqrt {2}} {2}}}}, S = {\ sqrt {2}} a ^ {2}.}$

Notes

↑ ¹ ² ³ Weisstein, Eric W. Star of Lakshmi on the Wolfram MathWorld website.

[mw-1] ¹ ² ³ Weisstein, Eric W. Star of Lakshmi on the Wolfram MathWorld website.

Lakshmi Star

Relationships

See also

Notes

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